reserve x,y,X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for PartFunc of C,COMPLEX;
reserve r1,r2,p1 for Real;
reserve r,q,cr1,cr2 for Complex;

theorem
  (f/g)(#)g = (f|dom(g^))
proof
A1: dom ((f/g)(#)g) = dom (f/g) /\ dom g by Th3
    .= dom f /\ (dom g \ g"{0c}) /\ dom g by Def1
    .= dom f /\ ((dom g \ g"{0c}) /\ dom g) by XBOOLE_1:16
    .= dom f /\ (dom (g^) /\ dom g) by Def2
    .= dom f /\ dom (g^) by Th6,XBOOLE_1:28
    .= dom (f|(dom (g^))) by RELAT_1:61;
  now
    let c;
    assume
A2: c in dom ((f/g)(#)g);
    then
A3: c in dom f /\ dom (g^) by A1,RELAT_1:61;
    then
A4: c in dom (f(#)(g^)) by Th3;
A5: c in dom (g^) by A3,XBOOLE_0:def 4;
    then
A6: g/.c <> 0c by Th8;
    thus ((f/g)(#)g)/.c = ((f/g)/.c) * (g/.c )by A2,Th3
      .= (f(#)(g^))/.c * (g/.c) by Th38
      .= ((f/.c)) *((g^)/.c) * (g/.c) by A4,Th3
      .= ((f/.c))*(g/.c)"*(g/.c) by A5,Def2
      .= ((f/.c))*((g/.c)" * (g/.c))
      .= ((f/.c))*1r by A6,COMPLEX1:def 4,XCMPLX_0:def 7
      .= (f|(dom (g^)))/.c by A1,A2,COMPLEX1:def 4,PARTFUN2:15;
  end;
  hence thesis by A1,PARTFUN2:1;
end;
